how to interpret EGARCH output in SAS ?

[srikanthyadav44]

Dear all

The SAS is using the following notation to run EGARCH

I run the EGARCH model on nifty50_returns in SAS with the following code 

proc autoreg data=garch;
model Nifty50_return= /
garch=(p=1,q=1, type=exp) maxit=1500;
run;

I got the following output.

I could not understand how to interpret the parameters. 

Which is the asymmetry parameter in the equation?

Earlier, i used to apply Eviews to run EGARCH. Eviews uses the following equation for EGARCH. 

LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) +
C(4)*RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))

For the same variable, I got the following output in Eviews

Variable	Coefficient	Std. Error	z-Statistic	Prob.
C	0.030646	0.015879	1.929991	0.0536
	Variance Equation
C(2)	-0.09955	0.01165	-8.54524	0
C(3)	0.127285	0.014562	8.740835	0
C(4)	-0.10378	0.006991	-14.8442	0
C(5)	0.974315	0.003433	283.8053	0

i am not able to correlate the results of views and GARCH and also unable to interpret the EGARCH parameters from SAS.

Can anybody help me in this regard?

Thanks in advance 

[SASCom1 · Accepted solution]

hello @srikanthyadav44 ,

The EGARCH equation used in PROC AUTOREG documentation follows that of the original Nelson(1991) paper 'Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica 59:347-70', and Hamilton(1994) 'Time Series Analysis' text book, page 668~669. Specifically, the g(z_t) equation in PROC AUTOREG documentation is the same as in equation (2.2) in Nelson(1991) paper on page 351, (with slight different parameter names in Hamilton(1994) equation). The equation for log(h_t) in PROC AUTOREG documentation is the same as equation (21.2.7) in Hamilton(again with slight different parameter names). For this specification, the parameter theta determines the asymmetric feature of the effect. When theta = 0, positive innovation(z_{t-1} > 0) has the same effect on volatility as a negative innovation(z_{t-1}<0) of the same magnitude. When theta not equal to zero, the effect is different for positive and negative innovation(z_{t-1} term). You may check Hamilton(1994) or Nelson(1991) for more detailed discussions.

I am not familiar with Eviews notation conventions, however, if the following equation you quoted:

LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(4)*RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))

means the following equation for the volatility:

log(h_t) = C(2) + C(3)*abs(residual_{t-1})/sqrt(h_{t-1})  + C(4)*(residual_{t-1}/sqrt(h_{t-1})) + C(5)*log(h_{t-1})                  (1)

then it seems that this equation (1) is a slight reparameterization of the equation used in PROC AUTOREG documentation and the Nelson(1991) and Hamilton(1994) discussed above, noting that z_t = residual_t/sqrt(h_t). For comparison purpose, I am writing the equation for log(h_t) in PROC AUTOREG documentation (using the same parameter names as the PROC AUTOREG output) below:

log(h_t) = EARCH0 + EARCH1*g(z_{t-1}  + EGARCH1*log(h_{t-1})             (2)

note that g(z_{t-1}) = theta*z_{t-1} + gamma*(abs(z_{t-1}) - E(abs(z_{t-1}))), where gamma = 1, E(abs(z_{t-1})) = sqrt(2/pi) for z_t~N(0,1), so the above equation (2) can be written as:

log(h_t) = EARCH0 + EARCH1*(theta*z_{t-1} + abs(z_{t-1}) - sqrt(2/pi)) + EGARCH1*log(h_{t-1})          

rearrange terms you can rewrite it as below in order to make it easier to compare to your Eviews equation (1) above:

log(h_t) = EARCH0 - EARCH1*sqrt(2/pi)

              + EARCH1*theta*z_{t-1} 

              + EARCH1*abs(z_{t-1})

              + EGARCH1*log(h_{t-1}                                                          (3)

Comparing equation (3) with equation (1) above, you can see the relationship between the two different parameterizations:

C(2) = EARCH0 - EARCH1*sqrt(2/pi) 

C(3) = EARCH1 

C(4) = EARCH1*theta

C(5) = EGARCH1

You may want to check your Eviews documentation to verify the above relations.